Implementation of target mass corrections and higher-twist effects in the xFitter framework

Knowledge of parton distribution functions (PDFs) at large momentum fraction $ x $ is not only important to study the flavor and spin dynamics of quarks in the nucleon, but also to search for signals of new physics at collider experiments. It is well known now that the nonperturbative QCD effects generally play a more significant role at such kinematic regions. In this work, we present an open-source QCD analysis of PDFs considering target mass corrections (TMCs) and higher-twist (HT) effects which are necessary when the analysis includes also data points from deep inelastic scattering (DIS) with larger values of $ x $ and smaller values of photon virtuality $ Q^2 $. To this aim, we use the xFitter package, as a comprehensive and powerful tool for studying PDFs. We report and discuss our recent activities on the implementation of TMCs and HT effects in the xFitter framework in a user-friendly way, so that the user can switch on or off them, and also choose between different options. We check the validity of the modifications by performing sample analyses of PDFs considering TMCs and HT effects and using a wide range of DIS data.


I. INTRODUCTION
Thanks to the high-energy scattering data from various accelerator facilities worldwide, significant advances have been made toward the internal structure of the nucleon using the framework provided by the quantum chromodynamics (QCD). To analyze this huge amount of data, concerted efforts are being made to improve information about the parton distribution functions (PDFs) of the proton [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]. Various groups are actively involved in extracting PDFs from experimental data, for instance JR [2], CTEQ [3,4], MMHT [5], NNPDF [6] and ABMP [7,8]. Although a wide range of experimental data is included in such analyses, the data with low W 2 (final state invariant mass) and Q 2 (hard scale) which are sensitive to the nonperturbative QCD effects are usually cut out, losing some sensitivity at high momentum fractions x. In return, as an example, one can refer to the analysis of the CTEQ-Jefferson Lab (CJ) collaboration [16] which has turned its attention to performing a global analysis of PDFs with the aim of maximally utilizing data from deep inelastic scattering (DIS) at the highest-x values. It is worth noting in this context that for extracting PDFs in the large-x region, it is crucial to consider all sources of corrections which may contribute to a comparable magnitude in the theoretical calculations, such as the target mass corrections (TMCs) and higher-twist (HT) effects [16][17][18][19][20]. To be more precise, these supplementary corrections become increasingly important as the hard scale Q 2 is lowered and x tends to 1.
Knowledge of PDFs at large x is important for several reasons. For example, accurate knowledge of the u and d quark distributions at high x is very important at collider experiments in searches for signals of new physics at high Q 2 values. On the other hand, the flavor and spin dynamics of quarks in the nucleon can be explored by studying the large-x region, considering this fact that the d/u ratio at large x is very sensitive to different mechanisms of spin-flavor symmetry breaking [21][22][23]. It should also be noted that the accurate recognition and calculation of the QCD backgrounds in new physics searches at hadron colliders would not be possible without precise determination of PDFs at large x.
Motivated by this need for a reliable and consistent determination of PDFs and their uncertainties using data at the highest-x values, we present in this work an open-source QCD analysis of PDFs considering TMCs and HT effects. In fact, the main goal of the present work is providing a user-friendly facility to perform global analysis of PDFs including TMCs and HT effects. To this aim, we use the xFitter package [24] and modify it to include related theoretical calculations and perform needed jobs correctly. Actually, implementation of TMCs and HT effects in the xFitter package, as a comprehensive and powerful tool for studying PDFs, can be of interest and also useful for further studies in this subject.
The contents of the present paper are as follows. In Sec. II, we discuss the theoretical framework which is used for calculating TMCs and HT effects on the DIS structure functions.
New developments implemented in the xFitter package and different options that user can choose are also explained. This section ends by introducing the fit framework we use in the present work as an example of PDFs analysis considering TMCs and HT effects utilizing the developed xFitter. In Sec. III, we briefly introduce the experimental data included in our sample analyses. Section IV is devoted to present and compare the results obtained from different analyses without and with considering TMCs and HT effects. In Sec. V, some cross-checks are presented in order to evaluate the validity of the calculations. Finally, we summarize our results and conclusions in Sec. VI.

II. THEORETICAL FRAMEWORK
It is well known now that considering TMCs and HT effects in any global analysis of PDFs including DIS data is an essential ingredient, if the analysis contains data points with larger values of x and smaller values of photon virtuality Q 2 . As mentioned in the Introduction, in this work, we are going to implement these corrections in the xFitter framework [24], so that the user can switch on or off TMCs and HT effects in a user-friendly way, and also choose between different options. In the next three subsections, we introduce the theoretical calculations of TMCs and HT effects which are implemented in xFitter and also our fit framework which is utilized in the present analysis.

A. Target mass corrections (TMCs)
Extracting PDFs, especially in the large-x region, requires some corrections arise from imposing exact kinematics on twist-2 matrix elements at finite values of 4-momentum transfer Q 2 . These corrections are related to x 2 M 2 /Q 2 , where M is the target mass. In fact, these target mass corrections are needed to consider the effect of mass for particles in the DIS experiment. To be more precise, in such a situation, the mass M modifies the scattering kinematics and therefore the DIS structure functions.
The TMCs were first calculated by Nachtmann [25] in the unpolarized case. It should also be noted that the same method has been applied to the polarized case in Ref. [26].
Accordingly, at finite Q 2 , the effects of the target and quark masses modify the Bjorken variable x with the light-front momentum fraction. For massless quarks, the parton lightfront fraction is given by the Nachtmann variable, where F 2,3,L are the structure functions in the M 2 /Q 2 → 0 limit and In addition to the OPE, there is an alternative formulation of TMCs [29,30] within the collinear factorization framework which implements TMCs directly in momentum space.
An important advantage of this approach is that it could also be applied to processes other than inclusive DIS [31,32], though it has been shown that two approaches are equivalent in O(1/Q 2 ). Overall, various prescriptions have been proposed so far for TMCs, using different approximations to the OPE and collinear factorization methods [28,[33][34][35][36]. There are also some efforts to implement TMCs in Mellin space through the contour integral representations both for the unpolarized [27] and polarized cases [37]. The latter also includes the corresponding relations for the twist-3 contributions to the polarized structure functions, though such contributions have also been derived in Ref. [38]. It should be noted that TMCs can also be considered in nonforward [39,40] and diffractive [41] scattering.
Since the evaluation of the convolution integrals in Eq. (2) can be time consuming, it is useful to have an approximate version of the target mass corrected structure functions.
Such approximations have also been presented in Ref. [28] which are in very good agreement with the exact results. For example, to achieve the approximate version of F TMC 2 in Eq. (2), the integral terms , respectively, which can be easily evaluated to obtain an upper bound for the contribution of the nonleading terms. In this way, the target mass corrected structure functions can be written in the following forms (please note that the related equation for F TMC L has not been presented explicitly in Ref. [28]): where µ is defined as µ = M 2 Q 2 . From a computational point of view, the approximate formulas of Eq. (4) can be calculated very fast, but calculation of the exact relations of Eq. (2) can be time consuming because they include also integral terms. The situation gets worse if the heavy flavor contributions F c,b 2,L to the structure functions should be calculated depending on the selected heavy quark scheme. Therefore, using the approximate TMCs can be a good option.
According to the above explanations, we have added two options for calculating TMCs to the xFitter package: "Approximate" which uses approximate formulas of Eq. (4) and "Exact" which uses exact relations of Eq. (2). However, since a version of TMCs has been implemented in the APFEL package [42] and, on the other hand, APFEL already exists in xFitter, it is also of interest to add TMCs option on xFitter according to the APFEL calculations. In fact, before the present work, it had not been used in xFitter by default, so that there was no any option for users to switch on or off TMCs. Consequently, at the present, we have modified xFitter to include TMCs in a user-friendly way so that the user can switch between three options: • Exact TMCs according to Eq. (2) • Approximate TMCs according to Eq. (4) • APFEL TMCs using formulas implemented in the APFEL package.
However, some points should be noticed. First, the xFitter calculations of the DIS structure functions, and then the reduced cross sections, have been classified according to the flavor number schemes, namely, zero-mass variable flavor number scheme (ZM-VFNS) [43], fixedflavor number scheme (FFNS) [44,45], ABM FFNS [46] RT [47][48][49], (S)ACOT [33,[50][51][52] and FONLL [53,54]. Note that the last three are a general-mass variable flavor number scheme (GM-VFNS). Since APFEL just works with FONLL scheme, for calculating TMCs using APFEL package, the user should choose FONLL and also DGLAP APFEL for theory type (collinear evolution with APFEL) in the "steering.txt" file of xFitter. Second, the user can choose any heavy flavor scheme if the Exact or Approximate TMCs is selected. Third, although using the Exact option for calculating TMCs is superior to the APFEL option due to abovementioned reasons, but APFEL has an advantage to perform related calculations up to 3 times faster. Fourth, APFEL uses a relation for F TMC 2 which has a less term than the relation presented in Eq. (2). To be more precise, the APFEL procedure can be considered equivalent to the Exact procedure, with a missing term in F TMC 2 . Nevertheless, as we checked it, using APFEL and Exact options leads to the results which are in excellent agreement with each other, though Exact leads to a somewhat better fit of DIS data than APFEL.

B. Higher twist (HT) effects
As mentioned in the Introduction, in addition to the TMCs, HT effects must also be taken into account in any analysis containing experimental data at low Q 2 and especially at large x. In the OPE framework, HT corrections to DIS processes are associated with matrix elements of operators involving multiquark or quark and gluon fields [55][56][57][58][59]. Although the details of these corrections are not yet fully understood, it is well known now that they are proportional to 1/Q 2 to the structure functions and become increasingly important as Q 2 is lowered and x tends to 1. Because dynamical higher twists involve nonperturbative multiparton interactions, there are theoretical difficulties of controlling power corrections in effective theories, so that it is difficult to quantify the shape of the higher twist terms from first principles. Consequently, HT effects are usually determined phenomenologically from experimental data. To be more precise, for extracting leading twist PDFs one can parametrize the HT contributions by a phenomenological form and fit the unknown parameters to the data [16,18,60,61]. It is also possible to extract the Q 2 dependence of HT contributions by fitting a phenomenological form in individual bins in x [62][63][64][65]. Moreover, the HT corrections can be obtained from the renormalon formalism [66]. It is worth noting in this context that considering the HT effects in the analysis of the diffractive PDFs can also lead to some improvements in the fit quality and change the shape of the extracted densities at some kinematic regions [67].
It should be mentioned that xFitter has already an option for considering HT according to GBW dipole model [68,69]. Although the dipole picture provides a natural description of the QCD reaction in the low x region, and then presents a good description of HERA data, it is not a suitable case for fixed target data. Consequently, in the present work, we are going also to add another option to xFitter for considering HT effects in a way that is appropriate for fixed target data. In this way, following the procedure used in the CJ15 analysis [16], we parametrize HT effects as a function of x and use the commonly used phenomenological form for the total structure function as follows: which includes higher twists but also other residual power corrections. In the above formula, F LT 2 is the leading twist structure function including TMCs and the function C HT is parametrized as where h 1 controls the rise of C HT at large x and h 2 allows for the possibility of negative HT at smaller x.
In the view of code, we have added another type of HT in xFitter, namely "CJ15", to already available type "Twist4" which is according to the GBW dipole model (without charm). Therefore, if a user uses the CJ15 option for "HiTwistType" in the "steering.txt" file, three parameters h 0 , h 1 and h 2 should be set as extra MINUIT [70] parameters. It should be noted that the twist4 option in xFitter considers the twist-4 contribution to neutral current (NC) reduced cross section, while the CJ15 option considers HT on the F 2 structure function.

C. Fit framework
As mentioned before, the main goal of the present work is providing a user-friendly facility to perform global analysis of PDFs including TMCs and HT effects by developing the opensource program xFitter [24]. In this way, in the present study, we just perform some simple QCD analyses of PDFs at next-to-next-to-leading order (NNLO) using developed xFitter to test the codes and check the results obtained. For sample analyses performed in the next section, we use the HERAPDF forms [9] for the generic parametrization of PDFs at the input scale µ 2 0 = 1.69 GeV 2 . It should be noted that the optimal parametrizations for PDFs can be found through a parametrization scan as described in [71]. In this way, the valence quarks xu v (x) and xd v (x), the antiquarks xŪ (x) = xū(x) and xD(x) = xd(x) + xs(x), and gluon xg(x) are parametrized as [9] xg( As usual, the normalization parameters A uv , A dv and A g are constrained by the well-known quark number and momentum sum rules. B parameters for sea quark distributions, namely BŪ and BD, are set to be equal, BŪ = BD. As well, constraint condition AŪ = AD(1 − f s ) is applied to ensure that xū → xd as x → 0. The strange-quark distribution is taken to be proportional to xD as xs = f s xD at µ 2 0 and the value of f s is chosen as f s = 0.4 [72][73][74]. It should be also noted that we impose kinematic cut Q 2 > 1.69 GeV 2 on all DIS data and  . II A). Finally, to take into account the effects of heavy quark masses, we choose the FONLL-C scheme as our analyses are performed at NNLO.

III. EXPERIMENTAL DATA
By introducing the theoretical framework in the previous section, in this section, we briefly introduce the experimental data which are used in our sample analyses. As mentioned before, since the main goal of the present work is developing xFitter to include TMCs and HT effects and the study of their impacts on the extracted PDFs, and not performing a comprehensive global analysis of PDFs, we just consider all available DIS data which are sensitive to TMCs and HT effects. In this way, in addition to the combined inclusive [9] and heavy-flavor production cross sections [75,76] at HERA and fixed target data from BCDMS collaboration [77] which are available in the xFitter package, we have also added and used fixed target data from SLAC [78], NMC [79], E665 [80] and Jefferson Lab [81].
All experimental datasets used in the present study have been listed in the first column of Table I.

IV. RESULTS
To study the impact of TMCs and HT effects on the extracted PDFs and quality of the fit, and also check the validity of the calculations, we perform three analyses considering an acceptable range of the DIS data introduced in the previous section. The first analysis we called "Base" is performed without considering TMCs and HT in a phenomenological framework introduced in Sec. II C. For the second analysis, namely "TMC", we include also TMCs with the formulation described in Sec. II A to study their impact on PDFs and the χ 2 value of individual datasets. Here we use the Exact option which leads to the most accurate result. It should be again noted that if one chooses the APFEL option, the same results are obtained though it leads to a somewhat worse fit of DIS data than Exact. Finally, in the third analysis, namely "TMC&HT", we consider also HT effects with formulation described in Sec. II B. Please note that for all analyses performed in this section we choose FONLL-C for the heavy quark scheme. Another point should be noted is concerning the rather poor fit quality for the JLab data with a χ 2 per point of χ 2 /N pts. ∼ 2 (the value of χ 2 is 292 for 142 data points) which is larger than the corresponding one obtained, e.g., from the CJ analysis [16]. show that considering TMCs solely can include the nonperturbative effects to a large extent and then HT helps to improve results even further. We found that such a situation is also established if one first turns on HT and then includes TMCs, which indicates that considering HT effects solely can also include the nonperturbative effects to a large extent.
However, the best option is considering TMCs and HT effects simultaneously. Table II   d v (x) to u v (x) for three analyses. One can clearly see an enhancement and reduction at medium and small x regions, respectively, which can be attributed to the inclusion of the TMCs and HT for the first one and quark number sum rules for the second one.
To investigate the impact of TMCs and HT on the extracted PDFs at higher values of Q 2 , as an example, in Fig. 2 we have shown the ratio of distributions obtained from three analyses Base, TMC and TMC&HT at Q 2 = 10 GeV 2 to those obtained from the Base analysis as a reference fit, so that the bands represent the related uncertainties. Actually, we have presented here these results as ratio plots to study the impact of TMCs and HT effects with more details. As can be seen, considering TMCs and Actually, our motivation comes from the fact that according to the results presented in the previous section (see Figs. 1 and 2), there are significant changes between the Base and TMC (TMC&HT) analyses, especially for the up valence distribution, which may be inconsistent at first glance. To understand this point further, it is useful to perform a fit to the same datasets, but applying a more restrictive cut on W 2 (in line with that commonly applied by global fitters) to remove the data points that are sensitive to the TMCs and HT effects (we do not impose any new cut on Q 2 and leave it as before as Q 2 > 1.69 GeV 2 ). In this way, the result of the Base analysis should be more consistent within errors with the TMC (TMC&HT) case.
In this regard, we first impose a new cut as W 2 > 15 GeV 2 on the data points for the TMC and TMC&HT analyses and compare the results obtained with the corresponding ones from the previous Base analysis with less restrictive cut W 2 > 3 GeV 2 . Actually, we expect that by removing the data points that are sensitive to the TMCs and HT effects from the TMC and TMC&HT analyses their impact on the extracted PDFs is neutralized. Figure 3 shows a comparison between the Base analysis with a less restrictive cut W 2 > 3 GeV 2 and the TMC and TMC&HT analyses with more restrictive cut W 2 > 15 GeV 2 . As can be seen, in this case, the result of the Base analysis is more consistent within errors with the TMC and TMC&HT analyses compared with the previous case (see Fig. 1). Therefore, one can conclude that the results presented in the previous section that show significant changes between the Base and TMC (TMC&HT) cases are admissible.
As another study, this time we perform a new Base analysis with a more restrictive cut W 2 > 15 GeV 2 imposed on data points and compare its results with the corresponding ones from the TMC and TMC&HT analyses with a less restrictive cut W 2 > 3 GeV 2 obtained in the previous section. Again we expect that the result of the Base analysis will be more consistent within errors with the TMC and TMC&HT analyses, but with larger errors. Figure 4 shows a comparison between the Base analysis with a more restrictive cut W 2 > 15 GeV 2 and the TMC and TMC&HT analyses with a less restrictive cut  For the case of TMCs, we have added three options to the xFitter package: "Approximate," 'Exact" and "APFEL." The APFEL TMCs just works with the FONLL heavy quark scheme, while the user can choose any heavy flavor scheme if the Exact or Approximate TMC is selected. However, APFEL has an advantage to perform related calculations up to 3 times faster than Exact TMCs. As we checked it, using APFEL and Exact options leads to the results which are in an excellent agreement with each other, though Exact leads to a somewhat better fit of DIS data than APFEL. For the case of HT, we have added the "CJ15" option to the xFitter package which parametrizes HT effects as a function of x and uses a commonly used phenomenological form for the total structure function.
We checked the validity of the modifications by performing sample analyses of PDFs with and without considering TMCs and HT effects and using a wide range of DIS data.
We affirm that considering such corrections can significantly change the extracted PDFs, especially the valence distributions, and improve the agreement between the theory and experiment. To be more precise, the u v (x) and d v (x) valence distributions are more affected by including TMCs and HT, since they are dominant at larger values of x than the others where TMCs and HT effects are also effective. The developed xFitter package including     The bands represent the related uncertainties. Note that the results have been plotted as ratios to the Base analysis. and d v (x)/u v (x) distributions at the initial scale Q 2 0 = 1.69 GeV 2 obtained from the Base analysis with less restrictive cut W 2 > 3 GeV 2 and the TMC and TMC&HT analyses with more restrictive cut and d v (x)/u v (x) distributions at the initial scale Q 2 0 = 1.69 GeV 2 obtained from the Base analysis with more restrictive cut W 2 > 15 GeV 2 and the TMC and TMC&HT analyses with less restrictive cut